Coding method and use of a receiver with a convolutional decoder

ABSTRACT

The invention relates to a coding method for modulating carrier signals with 16 different digital states (4 bit signals). The aim of the invention is to obtain a high synchronicity robustness and an at least partially improved coding gain. The coding parameters are obtained by the following steps: a) using a 2ASK/8PSK coding; b) choosing a convolutional code and determining all possible code word sequences with the free distance of the convolutional code; c) producing possible mappings by allocating a partial quantity of the 2ASK/8PSK channel bits to subsets; d) choosing the mapping at which after determination of the optimum radii (ac,c) of the two amplitudes for every possible mapping, the resulting minimum Euclidean distance takes a maximum value between two possible subset sequences code word sequence.

The invention concerns a coding method for modulation of carrier signalswith 16 different digital states (4-bit signals).

The invention also concerns the use of a receiver with a convolutionaldecoder, whose coding parameters are determined according to the codingmethod.

PRIOR ART

A 16-PSK coding is used, in particular, for transmission of digitalsignals, in which 16 different phase positions, i.e., phase differencesof 22.5°, must be detectable. 16 different digital states, i.e., 4-bitwords, can therefore be transmitted with phase information in thismethod.

The transmission link under practical conditions is subject to notinsignificant disturbances. The task in the receiver is therefore toselect all possible undisturbed receiving sequences that are to betransmitted with the highest probability through the disturbed receivingsequence. Calculation of the highest probability occurs in that theminimal Euclidean distance between the possible undisturbed receivingsequences and the actually observed disturbed receiving sequence isdetermined. If all receiving sequences are conceivable, in principle, anunduly high calculation demand develops for practical applications. Itis therefore known to carry out the calculation of the highestprobability recursively in the receiver and separate most of thesequences as not possible during the recursion. For this purpose, analgorithm offered by Viterbi is used for decoding of the convolutioncodes, so that the receiver is equipped with a so-called Viterbidecoder. Limitation of calculation of possible code word sequencesoccurs by determination of possible state transitions by means of atrellis diagram.

The use of a code that distinguishes 16 digital states is advantageousfor the transmittable data density and therefore the transmission speed.During use of a 16-PSK code, however, the problem of reducedsynchronization robustness arises, since relatively small phasedifferences must be detectable in the receiver. Methods are known fromDE 195 29 982 A1 and 195 29 983 A1 for increasing the synchronizationrobustness for the TC 16-PSK transmission in that, during the trackingphase, evaluation of the so-called best state information of the Viterbidecoder on the receiving side is carried out. Because of this, roughlythe synchronization robustness as in an 8-PSK constellation is achievedwith this type of transmission

A decisive quality criterion is also the “coding gain”, which can beachieved with reference to an uncoded transmission.

ADVANTAGES OF THE INVENTION

The underlying problem of the invention is to permit improvements withrespect to synchronization robustness and coding gain with a new codingmethod.

With this problem as point of departure, according to the invention. ina coding method of the type just mentioned, its coding parameters areobtained with the following process steps:

-   a) Use of a 2ASK/8PSK coding-   b) Selection of a deconvolution code and determination of all    possible code word sequences with the free distance of the    deconvolution code-   c) Formation of possible mappings by division of a partial set of    2ASK/8PSK channel bits into subsets-   d) Selection of those mappings, in which, after determination of the    optimal radii (ac, c) of the two amplitudes for each possible    mapping, the resulting minimal Euclidean distance between two    possible subset sequences (code word sequence) becomes maximal.

The invention therefore offers a dimensioning method, through which theminimal Euclidean distances are maximized, so that an increased codinggain is achieved for the 2 ASK/8 PSK coding employed here. Since an evenhigher coding gain is achieved according to the invention forpracticable application with the 2 ASK/8 PSK coding according to theinvention than with the ordinarily employed TC 16-PSK coding, the codingmethod according to the invention is superior. Moreover, the mentionedimprovement in synchronization robustness according to DE 195 29 982 A1and DE 195 29 983 A1 can also be applied in the method according to theinvention, in which case roughly the synchronization robustness of 4 PSKis achieved.

The invention offers the further advantage that ordinary receivers witha Viterbi decoder can be used, which need only be slightly altered fortheir use to receive coded signals according to the invention. Merely anadjustment of the so-called branch-metric table is generally required.

It has been found that the FM sensitivity during reception of codedsignals according to the invention can be improved by about 3 dBrelative to signals coded according to TC 16-PSK.

DRAWINGS

The invention will be further explained below with reference to theaccompanying drawing. In the drawing:

FIG. 1 shows a 2 ASK/8 PSK symbol constellation according to theinvention with 4 subsets that are formed by identical 2 LSB

FIG. 2 shows a depiction of the bit error rate as a function of thedisturbance spacing Eb/Nb, referred to one bit.

DESCRIPTION OF PRACTICAL EXAMPLES

The invention is explained on the example of a special deconvolutioncode (pragmatic trellis coding). FIG. 1 shows a 2 ASK/8 PSK symbolconstellation, as employable according to the invention. The allocationinto subsets A, B, C, D, formed by consistent 2 LSB (least significantbits), is arbitrary, in principle. The mapping shown in FIG. 1 isoptimal for special implementation in a pragmatic trellis coding.

Ordinarily, a deconvolution code is now chosen. For this deconvolutioncode according to the invention, all code word sequences are determinedwith the free distance of the deconvolution codes. It is assumed thatCWMAX such code word sequences exist. The running index is designatedCW=0 . . . CWMAX−1.

For design of the 2 ASK/8 PSK symbol constellation according to theinvention, the minimal Euclidean distance of the trellis-coded symbolalphabet is calculated as a gauge for the asymptotic error behavior.This minimal distance is calculated according to the formulad _(min)=min(d ^(code) _(min) , d ^(uncode) _(min))

The left part in parentheses considers the minimal Euclidean distance ofthe coded information bits. This term is determined by the selectedmapping and the proper use of the employed deconvolution code. The rightpart considers the minimal Euclidean distance in the partition of thethird plane (4*4 points), i.e., the minimal distance of the twounprotected information bits. This quantity can be obtained directlyfrom geometric calculations.

The deconvolution code usually employed in pragmatic trellis coding(generator polynomial (octal): 133, 171) has a free distance of 10. Forcalculation of the minimal distance of the coded information bit, theminimal Euclidean distance of the code word sequence must be determinedwith the output weight 10 from the code word sequence with the weight 0(i.e., the permanent null sequence).

The following table shows all possible 11 code word sequences withoutput weight 10. Input weight Code word sequence 1 1 (1.1) (1.0) (0.0)(1.1) (1.1) (0.1) (1.1) 2 2 (1.1) (0.1) (1.0) (1.1) (0.0) (1.0) (1.0)(1.1) 3 3 (1.1) (0.1) (0.1) (0.1) (0.1) (0.0) (0.1) (0.1) (1.0) (1.1) 42 (1.1) (1.0) (0.0) (0.0) (0.1) (0.1) (0.0) (1.1) (0.1) (1.1) 5 3 (1.1)(0.1) (1.0) (0.0) (1.0) (1.0) (0.1) (0.0) (0.1) (1.1) 6 3 (1.1) (1.0)(1.1) (0.1) (0.0) (0.0) (0.0) (1.0) (0.0) (0.1) (1.1) 7 3 (1.1) (1.0)(0.0) (0.0) (1.0) (1.1) (0.0) (0.0) (1.0) (1.0) (1.1) 8 4 (1.1) (1.0)(0.0) (0.0) (1.0) (0.0) (1.0) (0.0) (0.1) (0.1) (1.0) (1.1) 9 5 (1.1)(0.1) (0.1) (0.1) (0.0) (1.0) (0.0) (0.0) (0.0) (0.0) (1.0) (1.0) (1.1)10 4 (1.1) (1.0) (0.0) (0.0) (0.1) (1.0) (1.0) (0.0) (0.0) (0.0) (1.0)(0.0) (0.1) (1.1) 11 6 (1.1) (1.0) (0.0) (0.0) (1.0) (0.0) (1.0) (0.0)(1.0) (0.0) (0.0) (0.0) (0.0) (1.0) (1.0) (1.1)

The Euclidean distance of the code word sequence is calculated fromd _(Pfad)={square root}{square root over (n.d ₁₁ ² +m.d ₀₁ ² +k.d ₁₀ ²)}in which

-   -   d₁₁ denotes the minimal geometric distance of a 00→11 transition        and n denotes the number of these transitions in the considered        code word sequence    -   d₀₁ denotes the minimal geometric distance of the 00→01        transition and m denotes the number of these transitions in the        considered code word sequence    -   d₁₀ denotes the minimal geometric distance of the 00→10        transition and k denotes the number of these transitions in the        considered code word sequence.

This distance is therefore dependent on the selected mapping and theproperties of the deconvolution code.

A division of the 2 LSBs into subsets A, B, C and D is carried out forchoice of the mapping.

Overall, there are 24 mapping allocations that are shown in thefollowing table. The running index map therefore runs between 0 and 23.map Subset A Subset B Subset C Subset D 0 00 01 10 11 1 00 01 11 10 2 0010 01 11 3 00 10 11 01 4 00 11 01 10 5 00 11 10 01 6 01 00 10 11 7 01 0011 10 8 01 10 00 11 9 01 10 11 00 10 01 11 00 10 11 01 11 10 00 12 10 0001 11 13 10 00 11 01 14 10 01 00 11 15 10 01 11 00 16 10 11 00 01 17 1011 01 00 18 11 00 01 10 19 11 00 10 01 20 11 01 00 10 21 11 01 10 00 2211 10 00 01 23 11 10 01 00

The allocation of 2 MSBs is irrelevant for the dimensioning describedhere.

For the selected deconvolution code in a selected mapping map, theoptimal dimensioning is carried out. Generally, this is to be carriedout for all 24 possible allocations according to the table. The optimalsolution viewed overall is then the mapping, in which the resultingminimal Euclidean distance is maximal after determination of the radiiaccording to the described method.

The following apply as dimensioning equations:

-   1. Normalization: (average transmission output=1) 8 (ac)²+8c²=16-   2.    ${d\quad\frac{cod}{\min}} = {\left. {d\frac{uncod}{\min}}\Rightarrow{{{n \cdot d}\quad\frac{2}{11}} + {{m \cdot d}\quad\frac{2}{01}} + {{k \cdot d}\quad\frac{2}{10}}} \right. = \left( {2a\quad c} \right)^{2}}$    ${{if}\quad 2a\quad c} \leq {{\sqrt{c}}^{2} + {\left( {a\quad c} \right)^{2}\quad\left( {a \leq 0.577} \right)}}$    ${{{or}\quad{n \cdot {- d}}\quad\frac{2}{11}} + {{m \cdot d}\quad\frac{2}{01}} + {{k \cdot d}\quad\frac{2}{10}}} = {c^{2} + \left( {a\quad c} \right)^{2}}$    ${{if}\quad 2a\quad c} \geq {\sqrt{c^{2} + \left( {a\quad c} \right)^{2}}\quad\left( {a \geq 0.577} \right)}$

The already explained comments then apply for n, m and k, as well asd₁₁, d₀₁ and d₁₀. The minimal geometric distances d_(xx) are expressedby means of the parameters a and c, with a ≦1, so that equations 1 and 2represent the determination equations for a and c. These determinationequations are to be evaluated for all code word sequences CW=0 . . .CWMAX−1 with the free distance of the employed convolution code and all24 different allocations map=0 . . . 23.

For all value pairs (a(cw, map), c(cw, map)), the value pair(a(cw_(opt), map_(opt)), c(cw_(opt), map_(opt))) that gives the maximumvalues for the pairs (d^(code) _(min)(cw, map) d^(uncode) _(min)(cw,map)) for all code word sequences with the free distance of the employeddeconvolution code is to be used. It is then checked beforehand whetherthe minimal Euclidean distances d₁₁, d₁₀ and d₀₁ used in thedetermination equation also actually represent the corresponding minimaldistance for the value pair (a(cw_(opt), map_(opt)), c(cw_(opt),map_(opt))) determined from it.

If the mapping according to FIG. 1 is chosen, the dimensioning equationsare obtained as follows:

-   1. Normalization: (average transmission output=1) 8 (ac)²+8c²=16-   2.    ${d\quad\frac{cod}{\min}} = {\left. {d\frac{uncod}{\min}}\Rightarrow{{{n \cdot d}\quad\frac{2}{11}} + {{\left( {m + k} \right) \cdot d}\quad\frac{2}{01/10}}} \right. = \left( {2a\quad c} \right)^{2}}$    ${{if}\quad 2a\quad c} \leq {{\sqrt{c}}^{2} + {\left( {a\quad c} \right)^{2}\quad\left( {a \leq 0.577} \right)}}$    ${{{or}\quad{n \cdot d}\quad\frac{2}{11}} + {{\left( {m + k} \right) \cdot d}\quad\frac{2}{01/10}}} = {{c^{2} + {\left( {a\quad c} \right)^{2}{if}\quad 2a\quad c}} \geq {{\sqrt{c}}^{2} + {\left( {a\quad c} \right)^{2}\quad\left( {a \geq 0.577} \right)}}}$    with d₁₁=d₁=c−ac if c−ac≦ac{square root}{square root over (2)}    (a≧0.4142) or d₁₁=d₁=ac{square root}{square root over (2)} if    ac{square root}{square root over (2)}≦c−ac (a≦0.4142) and    d₀₁=d₁₀=d₀=ac{square root}{square root over (2(1−0.5))}.

The so described dimensioning of the 2 ASK/8 PSK symbol constellationrepresents the general case. In practice, a simplified dimensioningsuggests itself. For this purpose, it can be assumed that d₁=d₁₁≦{squareroot}{square root over (2)}.d₀=2.d₀₁={square root}{square root over(2)}.d₁₀. In this case, the code word sequence with the maximumn=n_(max) (i.e., the code word sequence with the maximum number of codewords 11) is the “worst case sequence”, which ultimately determines theparameters a and c. The parameters a and c were only determined for thiscase (a=0.5432, c=1.24270, d₁=0.56765 and d₀=0.56765; i.e., d₁≦{squareroot}{square root over (2)}.d₀) and the Euclidean distances determinedfor all code word sequences. As expected, the code word sequence withn=n_(max) determines the minimal distance d^(code) _(min), which isequal to d^(uncode) _(min) according to the design criterion. Evaluationof the dimensioning equations for all other code sequences givesd^(code) _(min)=d^(uncode) _(min) for the considered code word sequence.For the code word sequence with n=n_(max), d^(code) _(min)≠d^(uncode)_(min) and the minimum of it (min(d^(code) _(min).d^(uncode) _(min)) issmaller than during special calculation of the value pair (a . c) forthis code word sequence.

With the values determined above, the minimal distance in the partitionof the third plane is d^(uncode) _(min)=2ac=1.350.

For comparison of the Euclidean distances between the ordinary 16 PSKand the 2 ASK/8 PSK configuration according to the invention, theminimal Euclidean distances are summarized in the following table forthe code word transitions 00→11 and 00→01/10, as well as betweenneighboring symbols of the third partition step. Euclidean distance 2ASK/8 PSK 16 PSK d₀ = d₁₁ 0.5166 0.3902 d₁ = d₁₀ = d₀₁ 0.5677 0.7654$d_{2} = {d\frac{uncod}{\min}}$ 1.3501 1.4142Asymptotic Coding Gain

For the constellation dimensioned beforehand, the Euclidean distance isnow calculated for each code sequence with output weight 10 and comparedwith TC 16-PSK.

The following apply

for TC 16-PSK: d₁=d₁₁=0.7654, d₀=d₁₁=d₁₀=0.3902

for 2 ASK/8 PSK: d₁=d₁₁=0.5677, d₀=d₀₁=d₁₀=0.5167 Distance for Codesequence n m k TC16-PSK Distance for 2 ASK/8 PSK 0 4 1 1 1.627 1.350 1 31 3 1.538 1.426 2 2 5 1 1.444 1.499 3 3 1 3 1.538 1.426 4 2 3 3 1.4441.499 5 3 2 2 1.538 1.426 6 3 0 4 1.538 1.426 7 2 2 4 1.444 1.499 8 2 33 1.444 1.499 9 2 2 4 1.444 1.499 10 2 0 6 1.444 1.499

It can be interpreted from the table thatd ^(code) _(min) =d ^(uncode) _(min)=1.350.

The asymptotic coding gain relative to TC 16-PSK is now:$G = {{{20 \cdot \lg}\quad\frac{d_{\min}^{{TC8} - {8{{PSK}/{ASK}}}}}{d_{\min}^{{TC16} - {PSK}}}} = {{{20 \cdot \lg}\quad\frac{1.350}{\sqrt{2}}} = {{- 0.4}\quad{dB}}}}$

This means that the new constellation is asymptotically poorer by 0.4 dBthan TC 16-PSK.

Error Rates

The trend of the bit error rates over E_(bit)/N₀ of the 2 ASK/8 PSKconstellation is shown in FIG. 2, in comparison with TC 16-PSK. Theemployed coding in both cases is the pragmatic trellis coding with thepreviously selected convolution code.

The following table documents the coding gains that can be achieved withthis constellation relative to TC 16-PSK. Coding gain of TC8-8-PSK/ASKBit error rate Relative to TC16-PSK 10² 1.9 dB 10³ 1.8 dB 10⁴ 1.5 dB 10⁵1.2 dB 10⁶ 1.0 dB 10⁷ 0.6 dB Asymptotic −0.4 dB  

As in TC 16-PSK operation, auxiliary signals for carrier derivation canalso be obtained in the constellation of FIG. 1 for 2 ASK/8 PSK byevaluation of the best-state metric of the Viterbi decoder. Theauxiliary signals indicate whether the two LSBs in the received signalare equal or unequal. This information is sufficient in the presentmapping to decide whether the received signal point is rotated by an oddmultiple of 45° or a multiple of 90° relative to the reference carrier.Roughly the robustness of a QPSK-constellation can accordingly beachieved for the constellation containing 16 symbols of FIG. 1.

The described practical example shows that a coding gain between 1.0 and2.0 dB relative to the prior art TC 16-PSK can be achieved with thecoding according to the invention in the error rate range from 10⁻⁶ to10⁻².

The asymptotic loss of 0.4 dB is not relevant for practical systems thatare operated in the error rate ranges between 10⁻⁸ and 10⁻⁴.

The synchronization for the simple constellation according to theinvention is more robust than in TC 16-PSK, since only 8 permittedangles exist in the complex planes. As in TC 16-PSK, roughly thesynchronization robustness as in a QPSK constellation can be achieved inthe method according to the invention during the tracking phase byevaluation of so-called best state information of the Viterbi decoder onthe receiving side.

1. Coding method for modulation of carrier signals with 16 differentdigital states (4-bit signals), whose coding parameters are obtainedwith the following process steps: a) Use of a 2ASK/8PSK coding b)Selection of a deconvolution code and determination of all possible codeword sequences with the free distance of the deconvolution code c)Formation of possible mappings by allocation of a partial set of2ASK/8PSK channel bits into subsets d) Selection of the mappings, inwhich, after determination of the optimal radii (ac, c) of the twoamplitudes for each possible mapping, the minimal Euclidean distanceproduced between two possible subset sequences—code wordsequence—becomes maximal.
 2. Coding method according to claim 1,characterized by the fact that the following determination equations areused to determine the optimal radii (ac, c):8(ac)²+8c ²=16n·d ₁₁ ² +m·d ₀₁ ² +k·d ₁₀ ²=(2ac)²2ac≦{square root}{square root over (c ² +(ac) ² )}forn·d ₁₁ ² +m·d ₀₁ ² +k·d ₁₀ ² =c ²+(ac)²2ac≧{square root}{square root over (c²+(ac)²)},for in which d₁₁ is the minimal geometric distance of a subsettransition 00→11 and n is the number of these transitions in theconsidered code word sequence, d₀₁ is the minimal geometric distance ofa subset transition 00→01 and m the number of these transitions in theconsidered code word sequence and d₁₀ is the minimal geometric distancea subset transition 00→10 and k the number of these transitions in theconsidered code word sequence.
 3. Coding method according to claim 1,characterized by the fact that the subsets (A, B, C, D) are chosen sothat the following apply:d₁₁ =d ₁ =c−acfor c−ac≦ac·{square root}{square root over (2)}or d ₁₁ =d ₁ =ac·{square root}{square root over (2)}for ac·{square root}{square root over (2)}≦ c−acand d ₀₁ =d ₁₀ =d ₀ =ac{square root}{square root over ((1−0.5))}. 4.Coding method according to claim 3, characterized by the fact that, forcalculation of the optimal radii (ac, c), it is assumed forsimplification: d₁≦{square root}{square root over (2)}.d₀.
 5. Use of areceiver with a deconvolution decoder, especially a Viterbi decoder, forreceiving and decoding, whose coding parameters are determined accordingto the steps of the preceding claims.
 6. Use according to claim 5, inwhich derivation of the carrier is supported with auxiliary signals ofthe deconvolution decoder.